Aeolipile

Possibly the very first jet engine. Designed by Hero of Alexandria circa 100A.D., the aeolipile consisted of a boiler, two tubes extending from the top of the boiler which attached to opposite sides of a metal sphere, the sphere itself, and two additional bent tubes attached to the sphere which were left open on the ends.

Fill the boiler with water, light a fire beneath, steam rises up through the tubes attached to the sphere and ultimately escapes through the bent tubes, causing the sphere to spin.

What is an aeolipile?

And equally clearly, the aeolipile is pushed around by the reaction to emitted steam, so it's a reaction engine as well.

And because it's reacting to the steam jets, it's also a jet engine.

But what's a rocket-like jet engine?

It's a rocket.

Consider the fact that the propulsive force is created solely from the propellant that is held in a tank in the device. Consider also that the propellant is formed into a jet which pushes it around. Thus it can be seen that the aeolipile is a hot water rocket engine on a bearing.

Just putting a rocket on a bearing doesn't make it not a rocket, in the same way as putting a jet engine on a bearing doesn't make it not a jet engine. As a similar example, some experimental helicopters have used rocket tips; they're still rocket engines even though they go around in circles, with propellant fed from tanks in the body.

So if you agree that it's a jet engine, then it's also a rocket engine.

So by definition(*) the aeolipile was a rocket engine.

That means that rocket (engines) were invented first century A.D. in Alexandria, not in the 13th Century in China, as is commonly believed. (Of course rocket in the sense of a free flying vehicle or missile probably was invented in China, but that's a separate question).

So, it's a jet engine, and a steam engine and a reaction engine and a rocket engine... these things are not mutually exclusive...

Notes:

*- For the record the criteria for rocket engines are:

it takes the propellant from an attached tank (in an aeolipile it comes from the water tank)

it forms it into a jet which it uses for propulsion (in an aeolipile it comes out of nozzles and it is pushed around by the propulsive jet)